---
id: "2026-09-28-schiffer-pompeiu-convex-higher-dimensions"
url: "https://postcutoff.com/e/2026-09-28-schiffer-pompeiu-convex-higher-dimensions/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-09-28"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 3
editor: human editor
human_review: null
version: "2026-10-04"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-09-28-schiffer-pompeiu-convex-higher-dimensions/

# Convex counterexamples to Schiffer and Pompeiu in dimensions 3, 4, 6, 8, 10 and 14, made with Claude Opus 5.5 and GPT-6 Astra/Sol

On 28 Sep 2026 Jizhou Guo posted what he says are the first counterexamples to the Schiffer and Pompeiu conjectures in dimension three and higher. They are also the first convex counterexamples in any dimension: bounded convex non-ball domains in R^3, R^4, R^6, R^8, R^10 and R^14, with computer-assisted existence proofs. The paper says the work 'was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol)'.

## Key facts
- arXiv 2609.35419 (87 pages, v1 28 Sep 2026); supersedes the author's Zenodo preprint on dimensions four and six (doi 10.5281/zenodo.22999505)
- Dimensions 4, 6, 8, 10 and 14: adjoint-invariant domains in the rank-two compact Lie algebras u(2), so(4), su(3), so(5) and g2. Harish-Chandra's radial-part formula reduces each case to a planar Helmholtz problem, and Kostant's theorem reduces convexity to the Cartan section
- Dimension 3: an axisymmetric convex domain, plus a separate strictly star-shaped non-convex example
- Existence by Newton–Kantorovich contractions in weighted coefficient spaces: finite blocks checked in Arb ball arithmetic, infinite tails bounded analytically; verification code and certificates are in the arXiv ancillary files and on GitHub
- Acknowledgement, verbatim: 'This work was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol). The author takes full responsibility for the content of this paper.' The paper does not break down which parts each model did
- Builds on the August 2026 planar counterexamples of Colbrook–Stepaniants and Cao-Labora–de Dios Pont
- Status: single-author preprint, not peer-reviewed

## What happened

Eight weeks after the planar disproofs, a single author extended the counterexamples to higher dimensions and to convex domains. He used Lie-theoretic symmetry to reduce each case to a planar problem and certified the solutions with interval arithmetic.

## Why it matters

If it holds up, the Schiffer and Pompeiu conjectures fail even for convex domains, which was one of the natural fallback forms of the conjectures. It is also an example of a September 2026 paper that names the newest frontier models (Claude Opus 5.5 was released on 22 Sep) without saying what each one did.

## What is disputed or not yet verified

- Verification: Computer-assisted proof with public certificates; preprint, not peer-reviewed

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 151 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 90 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 181 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 120 days after its cutoff

## Sources

1. [arXiv 2609.35419: Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions three, four, six, eight, ten and fourteen](https://arxiv.org/abs/2609.35419) (arxiv.org, paper)
2. [GitHub: aster2024/schiffer-pompeiu-counterexamples (paper and verification)](https://github.com/aster2024/schiffer-pompeiu-counterexamples) (github.com, code)
3. [GitHub: aster2024/schiffer-pompeiu-r4 (earlier dimension-four version)](https://github.com/aster2024/schiffer-pompeiu-r4) (github.com, code)

## Changes

- 2026-09-30 (filed): Created

## Related

- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)
- 2026-09-22: [Anthropic releases Claude Opus 5.5](https://postcutoff.com/e/2026-09-22-claude-opus-5-5/index.md)
- 2026-09-03: [OpenAI releases GPT-6 Astra, its first GPT-6 model](https://postcutoff.com/e/2026-09-03-gpt-6-astra/index.md)
- 2026-08-03: [Planar Schiffer and Pompeiu conjectures disproved by two independent groups](https://postcutoff.com/e/2026-08-03-schiffer-pompeiu-conjectures-disproved/index.md)